This invention relates to a device for solving a symmetric linear system and, in particular, to a device for calculating, on a supercomputer having a high vector processing ability, an approximate solution of a partial differential equation by using a discretization method, such as a finite element method or a finite difference method.
A large sparce symmetric linear system arises from discretization of the partial differential equation on solving the partial differential equation by using the discretization method.
As well known in the art, a general symmetric linear system is given by: EQU Au=b, (1)
where A represents an N by N symmetric coefficient matrix with coefficient elements a.sub.ij (i, j=1, . . . , N), b, a right-hand side vector of Equation (1) with right-hand side elements b.sub.i, and u, a solution vector with solution elements u.sub.i. The symmetric coefficient matrix A is equal to a three-term sum of a diagonal matrix Da with diagonal elements a.sub.11, . . . , a.sub.NN of the coefficient elements plus an upper triangular matrix Ua with upper triangular elements of the coefficient elements plus a lower triangular matrix La with lower triangular elements of the coefficient elements. That is: EQU A=Ua+Da+La. (2)
Inasmuch as the symmetric coefficient matrix A is symmetrical, namely, a.sub.ji =a.sub.ji, one of the upper and the triangular matrices Ua and La is equal to a transpose matrix of the other of the upper and the trianglar matrices. That is: EQU Ua=La.sup.T, (3) EQU and EQU La=Ua.sup.T. (4)
Under the circumstances, a scalar processor calculates a numerical solution of the general symmetric linear system by a calculation method which uses the diagonal matrix Da and one of the upper and the lower triangular matrices Ua and La as input data for use in the symmetric coefficient matrix A.
However, the calculation method for use in a scalar processor is not available on a supercomputer having a high vector processing ability. This is because the calculation method for use in the scalar processor makes the supercomputer carry out calculations with shorter vector lengths. The supercomputer is, therefore, put into operation by a conventional calculation method which uses, as input data, all of the diagonal matrix Da, the upper triangular matrix Ua, the lower triangular matrix La, and the right-hand side vector b.
Therefore, the conventional calculation method for use in a supercomputer is disadvantageous in that it must prepare, as the input data for use in the symmetric coefficient matrix A, all of the coefficient elements of the symmetric coefficient matrix A. In addition, the conventional calculation method for use in the supercomputer requires a larger memory area for the input data.